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I have to give $5$ examples of $D_5$ acting on a set. So far, I have $D_5$ acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the $x$-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the $x$-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given?

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    $\begingroup$ Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. $\endgroup$ – Shaun Mar 8 at 14:12
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    $\begingroup$ @Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra". $\endgroup$ – Derek Holt Mar 8 at 14:27
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If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.

Let $\rho$ be an element of order $5$ in $D_5$ and let $\sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $\rho$ rotating the pentagon by $72^\circ$ and $\sigma$ reflecting it.

There are also actions corresponding to where $\rho$ rotates the pentagon by any other multiple of $72^\circ$ (including $0^\circ$).

There are also actions where $\rho$ rotates the pentagon but $\sigma$ does nothing to the pentagon.

You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).

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The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.

I think you might be good with this.

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    $\begingroup$ Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected. $\endgroup$ – Derek Holt Mar 8 at 13:38

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